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n-yoshinaga-2007INPC 1
Systematic studies of doublet bands in doubly-odd nuclei using a
simple model
N. N. Yoshinaga and K. HigashiyamaYoshinaga and K. HigashiyamaN. N. Yoshinaga and K. HigashiyamaYoshinaga and K. HigashiyamaDepartment of Physics, Saitama universityDepartment of Physics, Saitama university
Department of Physics, Chiba institute of technologyDepartment of Physics, Chiba institute of technology
Department of Physics, Saitama universityDepartment of Physics, Saitama university
Department of Physics, Chiba institute of technologyDepartment of Physics, Chiba institute of technology
Outline of my talkOutline of my talkExperiment studiesExperiment studies of doublet bands in A~130 of doublet bands in A~130 and our and our theoretical resultstheoretical results
A simple theoretical frameworkA simple theoretical framework and its application and its application to doubly-odd nucleito doubly-odd nuclei
Analysis of structure of doublet bandsAnalysis of structure of doublet bands
SummarySummary
Experimental studies of doubly-odd nuclei in A~130Experimental studies of doubly-odd nuclei in A~130
Yrast Yrast BandBand
Yrare BandYrare Band
T. Koike et al., Phys. Rev. C T. Koike et al., Phys. Rev. C 6363, 061304(R) , 061304(R) (2001). (2001).
11 2 11 2h h configuration
PraseodymiumPraseodymiumPraseodymiumPraseodymium
Truncated Shell model calculationsTruncated Shell model calculationsTruncated Shell model calculationsTruncated Shell model calculations
1010++1010++
1212++1212++
1414++1414++
1111++1111++
1313++1313++
1515++1515++
11/ 2 11/ 2h h
Neutron spinNeutron spin
Proton Proton spinspin
Core spinCore spin
0
1
2
3
10+
12+
14+
16+
11+
15+
9+
13+
15+
16+
12+
17+
11+
expt.
E(M
eV
)
17+
14+
10+
PTSM
13+
9+
8+
134La
Our results of the PTSM calculations Our results of the PTSM calculations (2005)(2005)The Pair-truncated shell model (PTSM) reproduces The Pair-truncated shell model (PTSM) reproduces
energy levels and electromagnetic properties of energy levels and electromagnetic properties of doublet bandsdoublet bands of doubly-odd nuclei with mass of doubly-odd nuclei with mass A~130.A~130.
The calculation becomes quite difficult because the The calculation becomes quite difficult because the configuration space grows up as the number of configuration space grows up as the number of valence nucleon increases.valence nucleon increases.
Band structure of doublet bands is well explained by Band structure of doublet bands is well explained by the movement of two spins ofthe movement of two spins of a neutron a neutron and and a protona proton, , weakly coupled with the weakly coupled with the even-even coreeven-even core. Thus the . Thus the PTSM provides us with an ideal tool for a study of PTSM provides us with an ideal tool for a study of doublet bandsdoublet bands..
We propose a very simple We propose a very simple model !model !
However…However…
Our Simple Model Our Simple Model Our Simple Model Our Simple Model
HamiltonianHamiltonian
Quadrupole coupling modelQuadrupole coupling modelBasis state of doubly-odd nucleusBasis state of doubly-odd nucleus
;; ,Ij jRL I R L
R
;j j L
,core c cH H H H H
,cor Re RRR R EH
RE
,cc c Q QH ,cc c Q QH H Q Q
: Collective core state (even-even : Collective core state (even-even nucleus)nucleus): Two-particle state (neutron and proton): Two-particle state (neutron and proton)
are fixed to describe even-even nucleus are fixed to describe even-even nucleus !!
1 3 5 7 9 11 13 15 17 19 21
0
1
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3
4
E(M
eV
)
I ( )
Theory
expt.
h
134La
11 2 11 2h h configuration
Theoretical energies and experimental energiesTheoretical energies and experimental energies
0
1
2
3
4
E(M
eV
)
130La
expt.
10+
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9+
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17+
19+
12+
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16+
11+
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Theory 8+
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11+
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19+
8+
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18+
9+
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15+
17+
19+
128La
expt. Theory
YrastYrareYrast
Yrare
YrastYrareYrast20+
(6+)
(8+)
(10+)
(12+)
(14+)
(16+)
(18+)
(20+)
(5+)(7+)
(9+)
(11+)
(13+)
(15+)
(17+)
(19+)
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Yrast and Yrare energies for 130La and 128LaYrast and Yrare energies for 130La and 128La
0
10
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30
40
11 13 15 17 190
10
20
30
40
11 13 15 17 19I h( )
B M
1(
;
I
I
1)
B E
2(
;
I
I
2)
[(
N
)2
]
eb
Theory
expt.
134La 132La
130La
128La
Staggering behavior of B(M1)/B(E2) ratiosStaggering behavior of B(M1)/B(E2) ratios
5N 7N
9N 11N
Analysis of QCM wave functionsAnalysis of QCM wave functions
RR
2 2cos
I j j I
I j I I j I
I
22 I IR R
Effective angle of neutron spin and proton Effective angle of neutron spin and proton spinspin
Square of core angular momentumSquare of core angular momentum
: Eigenstate obtained by the : Eigenstate obtained by the diagonalizationdiagonalization
Neutron spinNeutron spin
Proton Proton spinspin
Core spinCore spin
40
60
80
100
7 9 11 13 15 17 190
40
80
120
8 10 12 14 16 18 20
134La 128La
I h( )
Yrast
Yrare
(de
gre
e)
R2
h( 2
)
Angles between neutron and proton, and squares of core spinAngles between neutron and proton, and squares of core spin
5N 11N
Angles
Core spin
Structure of yrast statesStructure of yrast states
+
+
+
+
+
+
131
161
121
171
111+
141
101+
151
+191
+181
1111++1111++
1313++1313++
1515++1515++
1010++1010++
1212++1212++
1414++1414++
M1 transition
134La
Neutron spinNeutron spin
Proton Proton spinspin
Core spinCore spin
Structure of yrast statesStructure of yrast states
+
+
+
+
+
+
131
161
121
171
111+
141
101+
151
+191
+181
1111++1111++
1313++1313++
1515++1515++
1010++1010++
1212++1212++
1414++1414++
134La
Neutron spinNeutron spin
Proton Proton spinspin
Core spinCore spin Strong M1 transition
Structure of yrast statesStructure of yrast states
+
+
+
+
+
+
131
161
121
171
111+
141
101+
151
+191
+181
1111++1111++
1313++1313++
1515++1515++
1010++1010++
1212++1212++
1414++1414++
134La
Neutron spinNeutron spin
Proton Proton spinspin
Core spinCore spinWeak M1 transition
SummarySummarySummarySummaryWe propose a simple model (QCM) for We propose a simple model (QCM) for doublet bands in doubly-odd nuclei, doublet bands in doubly-odd nuclei, where where the neutron and the proton are the neutron and the proton are coupled with the core through coupled with the core through quadrupole interactions.quadrupole interactions.
The mechanism of the The mechanism of the strong staggering of strong staggering of B(M1)/B(E2) B(M1)/B(E2) ratiosratios is now well understood. It is now well understood. It explains why the strong staggering occurs explains why the strong staggering occurs only in the vibrational or transitional region, only in the vibrational or transitional region, and not in the deformed region.and not in the deformed region.
The model well reproduces energy spectra The model well reproduces energy spectra and electromagnetic properties of and electromagnetic properties of doublet doublet bandsbands..
Backups
Interaction strengthInteraction strength
More about our modelMore about our model
0.30 50 0.25 82 2.55 c Z N
0.10 50 0.05 82 1.35 c Z N
0.50 50 2.50 Z
Our model is different from the Our model is different from the particle-rotor particle-rotor modelmodel in in two respectstwo respects..
1.1. Information of rotor state is extracted Information of rotor state is extracted from experimental data of even-even from experimental data of even-even nucleus.nucleus.
2.2. All interactions are assumed to be of All interactions are assumed to be of quadrupole quadrupole types.types.
E2 operatorE2 operator
M1 operatorM1 operator
Electromagnetic transitionsElectromagnetic transitions
( 2) c cT E e Q e Q e Q
( 1) c s sT M g R g g s g g s
ce
e e
cg
g sg g sg : same values adopted in PTSM : same values adopted in PTSM calculationscalculations
: same values adopted in PTSM : same values adopted in PTSM calculationscalculations
: dipole moment of : dipole moment of even-even nucleuseven-even nucleus
: E2 transition of : E2 transition of even-even nucleuseven-even nucleus
+
+
+
+
+
+
+
+
+
+142
162
131
112
132
161
121
152
171
111+
141
101
+
+
122
91
1029282
81
+
+
+
+
+
+
11+10+9+8+
j j j j j j j j
151
123+
143+
71+
1.4
2.4
1.3
3.0
2.2
1.0
3.0
1.9
1.5
3.0
2.8
3.6
4.0
3.4
5.0
21
2215
17
20
168.6
19
7.012
7.2
12
5.95.7
11+
j j
+191
+181
172+
163+
182+
23
2423
22
19
1.4
2.53.0
2.4
6.7
0.50
E2 E2 transitiontransition
E2 E2 transitiontransition
M1 M1 transitiontransition
M1 M1 transitiontransition
Partial level scheme of Partial level scheme of 134134LaLaPartial level scheme of Partial level scheme of 134134LaLa
0
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E(M
eV)
132Cs 130Cs 128Cs 126CsYrast
YrareYrast
YrareYrast
YrareYrast
Yrare
8+9+10+
11+
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8+9+
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+10+
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+10+
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E(M
eV)
136Pr 134Pr 132Pr 130PrYrast
YrareYrast
Yrare
Yrast
Yrare
8+9+10+
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11 13 15 17 190
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0
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11 13 15 17 190
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11 13 15 17 19
132Cs
I h( )
I h( )
expt.
QCM
B
M1
(
;
130Cs
I
I
1)
128Cs 126Cs
136Pr 134Pr
132Pr 130Pr
B
E2
(
; I
I
2)
[(
N
)2 ]e
b